Third Lecture

• Clean determination of angles

... Bs → DsK or B → D(∗)π

... B → ππ with isospin analysis, etc.

• Factorization in B → D(∗)X decay

... How / why to test it

• Factorization in charmless decays

... different approaches

... some predictions / applications

• Summary / Conclusions

Angles — cleanly

B → ψKS,L — saw this before

• Clean measurement possible because |λψKS,L| − 1� 1

afCP =Γ[B0(t)→ f ]− Γ[B0(t)→ f ]Γ[B0(t)→ f ] + Γ[B0(t)→ f ]

= Sf sin(∆mt)− Cf cos(∆mt)

λfCP = qp

AfCPAfCP

= ηfCPqp

AfCP

AfCP, Sf = 2Imλf

1+|λf |2, Cf = 1−|λf |2

1+|λf |2

Tree: AT ∼ VcbV∗cs

Penguin: AP ∼[λ2]VtbV

∗ts f(mt) +

[λ2]VcbV

∗cs f(mc) +

[λ4]VubV

∗us f(mu)

= VcbV∗cs︸ ︷︷ ︸ [f(mc)−f(mt)]+VubV ∗us︸ ︷︷ ︸ [f(mu)−f(mt)]

same as Tree phase suppressed by λ2

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λψKS,L = ∓(V ∗tbVtdVtbV ∗td

)(VcbV

∗cs

V ∗cbVcs

)(VcsV

∗cd

V ∗csVcd

)= ∓e−2iβ ⇒ ImλψKS,L = ± sin 2β

Z L — SSI p.3/1

Bs→ ψφ and Bs→ ψη(′)

• Analog of B → ψKS in Bs decay — determines the phase between Bs mixingand b→ ccs decay, βs, as cleanly as the determination of β

βs is a small angle (of order λ2) inone of the “squashed” UT’s

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Without aψKWith aψK

sin2βS

CL

CK Mf i t t e r

(hep-ph/0202010)

• ψφ is a VV final state, so the asymmetry will be diluted by the CP -odd component⇒ A large asymmetry would clearly signal NP

ψη(′), on the other hand, is pure CP -even

Z L — SSI p.3/2

B → ππ — the problem

• There are tree and penguin amplitudes, just like for ψKS

“Tree” (b→ uud): AT ∼[λ3]

VubV∗ud

“Penguin”: AP ∼[λ3]VtbV

∗td f(mt)+

[λ3]VcbV

∗cd f(mc)+

[λ3]VubV

∗ud f(mu)

(unitarity) ∼[λ3]

VubV∗ud︸ ︷︷ ︸ [f(mu)− f(mt)] +

[λ3]VcbV

∗cd︸ ︷︷ ︸ [f(mc)− f(mt)]

same as Tree phase not suppressed

Two amplitudes with different weak- and possibly differentstrong phases; their values not known model independently

d

d

W

B

u

u

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d

b

π

π

+

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bu, c, t

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π

-π

+

Define P and T by: Aπ+π− = T (VubV ∗ud) + P (VcbV ∗cd)

Ratio of Kπ and ππ rates indicates |P/T | ∼ 0.2− 0.4, i.e., |P/T | 6� 1

• Possible solutions: (1) eliminate P ; or (2) attempt to calculate P

Z L — SSI p.3/3

B → ππ — isospin analysis

(u, d): I-spin doublet

other quarks and gluons: I = 0

γ, Z: mixtures of I = 0, 1

(ππ)`=0 → If = 0 or If = 2(1× 1) (∆I = 1

2) (∆I = 32)

I = 0 final state forbidden by Bose symmetry

Hamiltonian has two parts: ∆I = 12 ⇒ If = 0

Hamiltonian has two parts: ∆I = 32 ⇒ If = 2 ... only two amplitudes

3 rates: B0 → π+π−, B0 → π0π0, and B− → π0π− determine magnitudes andrelative phase of two amplitudes ... similarly for B0 and B+ decay

In practice, need all (tagged) rates + time dependent asymmetry in B → π+π−

Note: γ and Z penguins violate isospin and yield some (small) uncertainty

Z L — SSI p.3/4

Isospin analysis (cont.)

A+− ≡ A(B0 → π+π−), A+− ≡ A(B0 → π+π−),A00 ≡ A(B0 → π0π0), A00 ≡ A(B0 → π0π0),A+0 ≡ A(B+ → π+π0), A−0 ≡ A(B− → π−π0).

Isospin symmetry implies that 6 amplitudesform two triangles with a common base

1√2A+−+A00 = A+0,

1√2A+−+A00 = A−0

|A+0| = |A−0|

2δ = difference between arg λπ+π− and 2α

(constrained by any limit on π0π0 rate – later)

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(hep-ph/9712306)

B → ρπ: 4 isospin amplitudes⇒ pentagon relationsB → ρπ: Dalitz plot analysis would allow considering π+π−π0 final state only

Z L — SSI p.3/5

Implications of current data

• Babar and Belle measured:

Sπ+π− =2 Imλππ

1 + |λππ|2≡ sin 2αeff , Cπ+π− =

1− |λππ|2

1 + |λππ|2

λππ = e−2iβ e−iγ + P/T

eiγ + P/T

If P/T were small, then |λππ| ' 1 and Sπ+π− = Imλππ ' − sin 2(β + γ) = sin 2α

Cπ+π− measures: |λππ|2 =1− Cπ+π−

1 + Cπ+π−(note: S2 + C2 ≤ 1, and = 1 iff Reλ = 0)

Central values of Cπ+π− imply Babar: −0.30± 0.25± 0.04 ⇒ modest P/T

Central values of Cπ+π− imply Belle: −0.94+0.31−0.25 ± 0.09 ⇒ large P/T

• To extract α from Sπ+π− alone, need to know magnitude and phase of P/T

Z L — SSI p.3/6

Implications for P/T — another way

Assume the SM is correct, use Sππ and Cππ measurements to constrain magni-tude and phase of P/T (≡ zππ eiφππ)

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GR �

[2001]�

BBNS [2001]�

Constraint from SM(ρ,η� ) and Sππ, Cππ (BABAR) confidence level�

zπ� π�

φ ππ

(d

egre

es)

CK Mf i t t e r

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GR �

[2001]�

BBNS [2001]�

Constraint from SM(ρ,η� ) and Sππ, Cππ (Belle) confidence level�

zπ� π�

φ ππ

(d

egre

es)

CK Mf i t t e r

(Hocker @ FPCP)

need more data...

Z L — SSI p.3/7

Bounding α− αeff

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Standard SM fit

α (deg)

α eff

(de

g)

Constraint from GL isospin analysis (BABAR) confidence level�

CK Mf i t t e r

R&�

D

Isospin relations + branching ratios + limit on B → π0π0 ⇒ bound on α− αeff

No strong constraint from present bound on B → π0π0

Z L — SSI p.3/8

Bs→ D±s K∓ — a different story

• Interference between Bs and Bs decay — only one tree amplitude in each case

Four amplitudes: BsA1→ D+

s K− (b→ cus) , Bs

A2→ K+D−s (b→ ucs)

Four amplitudes: BsA1→ D−s K

+ (b→ cus) , BsA2→ K−D+

s (b→ ucs)

AD+s K−

AD+s K−

=A1

A2

(VcbV

∗us

V ∗ubVcs

),

AD−s K+

AD−s K+

=A2

A1

(VubV

∗cs

V ∗cbVus

)Relative strong phase and magnitudes of A1 and A2 are unknown, still theoryerror is eliminated if four time dependent rates are measured:

λD+s K−

λD−s K+ =(V ∗tbVtsVtbV ∗ts

)2(VcbV

∗us

V ∗ubVcs

)(VubV

∗cs

V ∗cbVus

)= e−2i(γ−2βs−βK)

• Similarly, Bd → D(∗)±π∓ determines γ + 2β: λD+π− λD−π+ = e−2i(γ+2β)

... ratio of amplitudes O(λ2)⇒ expected asymmetries are small

Z L — SSI p.3/9

B±→ (D0, D0)K±→ fiK±

• B± → K±D: theoretically clean, experimentally very hard (Gronau-Wyler)

|A(B+ → K+D0)||A(B+ → K+D0|

∼ λ

Nc

• B± → K±(D0, D0)→ K±fi (i = 1, 2) (Atwood, Dunietz, Soni)

make use of large final state interaction in D decay

Idea: B+ → K+D0 → K+fi in doubly Cabibbo suppressed D0 decay

Idea: B+ → K+D0 → K+fi in Cabibbo allowed D0 decay (e.g.: fi = K−π+/ρ+)

Need at least 2 final states Total Br’s ∼ 10−7 — statistics?

• Many ideas on the market would become a lot simpler if some of the hadronicdecay amplitudes were understood

Z L — SSI p.3/10

Factorization in b→ c

Factorization in b→ c exclusive decays

B D

πStart from OPE; estimate matrix elements of four-quark operators by grouping the quark fields intotwo that mediate B → D, and two that candescribe vacuum → π — Are gluons connectingB&D to π either calculable or power suppressed?

• “Naive” factorization: 〈Dπ|cbud|B〉 ∼ FB→D fπSince B and D are “soft” and π is “collinear”, “color transparency” provides aphysical picture for factorization (early 90’s: Bjorken; Dugan, Grinstein)

Configuration of brown muck in D changes only slightly, π is a fast color dipole

This picture expected to hold for B → D(∗)X, as long as EX/mX � 1

Cannot be the full story: Wilson coefficients (of cbud operators) scale dependent

Z L — SSI p.3/11

Factorization in b→ c (cont.)

• “Generalized” factorization: 〈Dπ|cbud|B〉 ∼ FB→D∫ 1

0dxT (x, µ)φπ(x, µ)fπ

(proposed: Politzer, Wise; 2-loop proof: Beneke, Buchalla, Neubert, Sachrajda; all orders proof: Bauer, Pirjol, Stewart)

Fully consistent formulation, scale and scheme dependence cancels order-by-order between Wilson coefficients Ci(µ) and matrix elements 〈Oi(µ)〉

No OPE — corrections presumed to beO(Λ/mb)n but this is not firmly established(Depends on details of B, D, π wave-functions)

Proof applies when meson that inherits the spectator quark from the B is heavyand the other is light

• Factorization also holds in the large number of colors limit(Nc → ∞ with αsNc = const.) in all B0 → M−1 M

+2 type

decays, corrections ∝ 1/N2c

(a) (b)

(c)

B0 M−1

Z L — SSI p.3/12

Factorization tests

• Factorization has been observed to work in B0 → D(∗)+ π− decays at the <∼ 10%level (in amplitudes) ...it gets really interesting just below this (∼ 1/N2

c )

Want to understand quantitatively accuracy of factorization in different processes

• ∼ 35% corrections for B− matrix elements have been observedSpectator in B going into π expected to be power suppressed

� � � �π− D0

B−

B0, B− D0, D+ π−

B(B− → D0π−)B(B0 → D+π−)

= 1 +O(ΛQCD/mc,b) , data: ∼ 1.8± 0.2

Ratio appears universal across channels (D/D∗, π/ρ)

Z L — SSI p.3/13

Isospin again: B → Dπ

• Classify amplitudes in terms of isospin (conserved by strong interaction) insteadof “tree” and “color suppressed tree”, etc.

Two isospin amplitudes for B decay to (Dπ) in If = 12 or If = 3

2

Three measurable rates ⇒ 1 relation:

A(B− → D0π−) = A(B0 → D+π−) +√

2A(B0 → D0π0)

• Three rates determine |A1/2|, |A3/2| and their relative strong phase

δ ∼ 30◦ (CLEO, Belle, Babar)

QCD factorization predicts δ ∼ O(ΛQCD/mc,b)

Not clear yet what sets the scale for the size of corrections

Z L — SSI p.3/14

Origin of factorization?

• The color transparency argument relies on M2 being fast (m/E � 1); the large-Nc argument is independent of this

Would be nice to observe deviations that clearly distinguish between expectations

– At the level of existing data, factorization also works in B → D(∗)s D(∗) when

both particles are heavy

– Check if factorization is worse in B0 → D(∗)−s π+ than in B0 → D(∗)+ π− ?

Need B → π form factor ↖ should be |Vub/Vcb|2× power suppressed

– “Designer mesons”: Study factorization breaking in decays that vanish in naivefactorization (so αs & 1/m corrections important), e.g, B0 → D(∗)+ a0/b1/π2

Rates at 10−6 level — soon accessible? (Diehl & Hiller)

Z L — SSI p.3/15

Factorization in B → D(∗)X

• Study accuracy as a function of the kinematics, with fixed multi-body final states

Expect some nonperturbative corrections to grow as mX increases (ZL, Luke, Wise)

Compare B → D∗4π with τ → 4π (allows 0.4 <∼ mX/EX <∼ 0.7)

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B decay data

Observing deviations that grow with mX would provideevidence that perturbative QCD is an important part ofthe success of factorization in B → D∗X

Different charge modescan disentangle back-grounds from D∗∗, etc.

(CLEO)

bW- u

d}d }D +

d

{Bo

*

(a)

bW- u

d}d c}D o

d

{Bo

*

(b)

b

W- ud

}d

}D o

d

{Bo*

(c)

c

c

Z L — SSI p.3/16

Factorization in charmless decays

Factorization in charmless B decays

• Two contributions:

B

π

π B π

π

Two proposals:(1) “QCD factorization:” (Beneke, Buchalla, Neubert, Sachrajda)

〈ππ|Oi|B〉 ∼ FB→π T (x)⊗ φπ(x) + T (ξ, x, y)⊗ φB(ξ)⊗ φπ(x)⊗ φπ(y)

– Sudakov suppression at the b mass scale is not effective in the endpointregions of quark distributions

– Two terms have same size in ΛQCD/mb power counting

– Second term suppressed by αs(mb)

– Small strong phases

Z L — SSI p.3/17

Factorization in charmless B decays

• Two contributions:

B

π

π B π

π

Two proposals:(2) “Perturbative QCD:” (Keum, Li, Sanda)

〈ππ|Oi|B〉 ∼ 0 + T (xi, bi)⊗ φB(x3, b3)⊗ φπ(x2, b2)⊗ φπ(x1, b1)

– Sudakov suppression effective in the regions xi ∼ ΛQCD/mb and 1/bi ∼ ΛQCD

– kT factorization, 1/bi ∼√

ΛQCDmb dominates

– Larger strong phases, annihilation & penguin contributions

Z L — SSI p.3/18

Main issues

Huge body of literature due to importance for CP violation

– Power counting depends on treatment of Sudakovs

BBNS: form factors are nonperturbative inputs

KLS: Sudakov suppression renders form factors calculable

– Some formally O(ΛQCD/mb) terms in BBNS are known to be large

Chirally enhanced terms: 2m2K/mbms ∼ 1

– Other issues raised:Charming penguins (Ciuchini et al.)

Intrinsic charm (Brodsky & Gardner)

Scale where π form factor becomes asymptotic ∼ x(1− x)

It very hard to test the assumptions ... need large variety of rates and direct CPV,for some of which the predictions differ

Z L — SSI p.3/19

B → hh rates vs. predictions

(Nir @ ICHEP)

Although the starting points, and predictions for direct CPV and phases in P/T

differ, both BBNS and KLS can reproduce these rates by now

Experimental data is crucial

Z L — SSI p.3/20

(ρ, η) using charmless rates

Inputs: B → ππ/Kπ rates + BBNS

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CK Mf i t t e r

R&D

Z L — SSI p.3/21

γ from Bd,s→ hh

• B → Kπ/ππ: Combination of rates — careful

Need some assumptions on (some of): rescattering effects, penguins, factoriza-tion, SU(3)

• Bd → π+π− vs. Bs → K+K−: see: F. Wurthwein’s talk tomorrow (Fleischer)

Idea: two decays related by u-spin, that exchanges d↔ s

Corrections to the u-spin limit are order ms/Λ, just as for SU(3)Need to constrain it somehow from other measurements

My feeling / hope: measure all possible Bd,s → ππ, Kπ, KK asymmetries andrates, we’ll figure out something, build a case...

Z L — SSI p.3/22

Summary — factorization

• In decays such as B0 → D(∗)+ π− factorization has become well established

• Some power suppressed corrections (formally order ΛQCD/mc) appear sizable

No evidence yet of factorization becoming a worse approximation in B → D(∗)X

as mX increases

• In charmless decays there are two approaches to factorization, BBNS and KLS

• Different assumptions and power counting, and sometimes different predictionsTesting the assumptions in a conclusive way does not seem easy

• Theoretical progress in understanding semileptonic form factors in the small q2

region may help to understand importance of Sudakov effects

• New and more precise data will be crucial to test factorization and tell us aboutsignificance of unknown power suppressed terms in various processes

Z L — SSI p.3/23

Summary

Final remarks

• To overconstrain CKM, all possible clean measurements are very important, bothCP violating and conserving, even if redundant in SM (correlations important)

• The key processes are those which give clean information on short distanceparameters ...one theoretically clean measurement is worth ten dirty ones

• It changes with time what is theoretically clean — significant recent progress for:

– Determination of |Vub| from inclusive B decay

– Exclusive rare & semileptonic form factors at small q2

– Factorization in certain nonleptonic decays

• Studying CKM/CPV and hadronic physics is complementary; except for a few veryclean cases several measurements needed to minimize theoretical uncertainties— data will help to get rid of nasty things hard to constrain otherwise

Z L — SSI p.3/24

A (near future & personal) best buy list

• |Vtd/Vts|: Tevatron should nail this, hopefully very soon (lattice caveats?)

• Rare decays: B → Xsγ near theory limited; q2 distribution in B → Xs`+`− will

be very interesting

• |Vub|: reaching <∼ 10% would be very significant (assumes understanding |Vcb|;a Babar/Belle measurement that may well survive LHCB/BTeV)

• β: reduce error in φKS, η′KS (and D(∗)D(∗)) modes

• βs: is CPV in Bs → ψφ small?

• α: how small is B → π0π0? How big are other resonances in ρ−π Dalitz plot?

• γ: clean modes hard, test SU(3) relations, factorization and other approaches

• try B → `ν, search for “null observables” [aCP (b→ sγ), etc.], for enhancementof B → `+`−, etc.

(apologies for omissions!)

Z L — SSI p.3/25

Conclusions

• The CKM picture is predictive and testable — it passed its first nontrivial testand is probably the dominant source of CPV in flavor changing processes

• The point is not only to measure the sides and angles of the unitarity triangle,(ρ, η) and (α, β, γ), but to probe CKM by overconstraining it in as many waysas possible (rare decays, correlations!)

• The program as a whole is a lot more interesting than any single measurement;all possible clean measurements are important, both CPV and CPC

• Many processes can give clean information on short distance physics, andthere is progress towards being able to model independently interpret manyinteresting observables

“This is not the end. It is not even the beginning of the end.

But it is, perhaps, the end of the beginning.”

— W. Churchill (Nov. 10, 1942)

Z L — SSI p.3/26